# Locally Quasi-Convex Compatible Topologies on a Topological Group

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem 1.1.**([2]) If a locally quasi-convex group G is Čech complete (in particular, complete metrizable or locally compact abelian (LCA)), then G is a Mackey group.

**Corollary 1.2.**If G is an LCA group, then $\left|\mathcal{C}\right(G\left)\right|=1$ if and only if G is compact.

**Questions 1.3.**Let G be a locally quasi-convex topological group.

- (a)
- [3] (Question 8.92) Compute the cardinality of the poset $\mathcal{C}\left(G\right)$.
- (b)
- [3] (Problem 8.93) Under which conditions on the group G is the poset $\mathcal{C}\left(G\right)$ a chain?

#### 1.1. Measuring Posets of Group Topologies

**Definition 1.4.**Two posets $(X,\le )$ and $(Y,\le )$ are:

- isomorphic (we write $X\cong Y$) if there exists a poset isomorphism $X\u27f6Y$;
- ([9,10,11]) quasi-isomorphic (we write $X\stackrel{q.i.}{\phantom{\rule{0.0pt}{0ex}}\cong}Y$) if there exist poset embeddings:$$(X,\le )\hookrightarrow (Y,\le )\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(Y,\le )\hookrightarrow (X,\le )$$

**Example 1.5.**Let X be an infinite set, and let ${\mathbf{Fil}}_{X}$ be the set of all free filters (i.e., filters $\mathcal{F}$ on X with $\bigcap \mathcal{F}=\varnothing $) ordered by inclusion. The bottom element ${\phi}_{0}$ of ${\mathbf{Fil}}_{X}$ is known as the Fréchet filter; its elements are the complements of finite sets. A filter on X is free iff it contains ${\phi}_{0}$. For the sake of completeness, we shall add to ${\mathbf{Fil}}_{X}$ also the power set $\mathcal{P}\left(X\right)$ of X to obtain the complete lattice:

**Proposition 1.6.**For every infinite group G, there exists a poset embedding $\mathcal{T}\left(G\right)\to {\mathfrak{F}}_{\left|G\right|}$.

**Proof.**Every Hausdorff group topology τ on G is completely determined by the filter ${\mathcal{N}}_{G,\tau}$ of all τ-neighborhoods of zero. Since τ is Hausdorff, zero is the only common point of all members of ${\mathcal{N}}_{G,\tau}$. Hence, by restricting this filter to the set $X=G\backslash \left\{0\right\}$, we obtain an element of ${\mathfrak{F}}_{X}$. In other words, we defined an injective monotone map $\mathcal{T}\left(G\right)\to {\mathfrak{F}}_{X}$. Note that the discrete topology on X is mapped to $\mathcal{P}\left(X\right)$. ☐

**Fact 1.7.**[2] $\mathcal{C}(G,\tau )$ is a dcpo, i.e., stable under directed suprema taken in the complete lattice $\mathcal{L}\left(G\right)$.

#### 1.2. Main Results

- (a)
- If G is a locally quasi-convex abelian group and K is a compact subgroup of G, then $\mathcal{C}\left(G\right)\cong \mathcal{C}(G/K)$ (Theorem 3.15).
- (b)
- If H is an open subgroup of G, then there exist poset embeddings $\mathcal{C}(H)\stackrel{\Psi}{\phantom{\rule{0.0pt}{0ex}}\hookrightarrow}\mathcal{C}(G)\stackrel{\Theta}{\phantom{\rule{0.0pt}{0ex}}\hookleftarrow}\mathcal{C}(G/H)$ (Theorem 3.6 and Corollary 3.11).
- (c)
- For every discrete group D of infinite rank, the set $\mathcal{C}\left(D\right)$ is quasi-isomorphic to the set of filters on D (Lemma 4.5).

**Corollary 1.8.**If a locally quasi-convex group $(G,\mathcal{T})$ has an open subgroup of infinite co-rank, then:

**Definition 1.9.**A topological abelian group G is called r-disconnected if G has an open subgroup H of infinite co-rank.

**Definition 1.10.**For a topological group G define the discrete rank (d-rank) of G by:

**Corollary 1.11.**If an LCA group G has $\varrho \left(G\right)\ge \mathfrak{c}$, then $\mathcal{C}\left(G\right)\stackrel{q.i.}{\phantom{\rule{0.0pt}{0ex}}\cong}{\mathfrak{F}}_{\varrho \left(G\right)}$.

**Corollary 1.12.**Under the assumption of CH, $\mathcal{C}\left(G\right)\stackrel{q.i.}{\phantom{\rule{0.0pt}{0ex}}\cong}{\mathfrak{F}}_{\varrho \left(G\right)}$ holds for every non-σ-compact LCA group.

**Notation and preliminaries.**We denote by $\mathbb{N}=\{0,1,2,\dots \}$ the natural numbers, by $\mathbb{P}$ the prime numbers, by $\mathbb{Z}$ the group of integers, by $\mathbb{Q}$ the group of rational numbers, by $\mathbb{R}$ the group of real numbers and by ${\mathbb{Z}}_{m}$ the cyclic group of order m. Let $\mathbb{T}$ denote the quotient group $\mathbb{R}/\mathbb{Z}$. We shall identify it with the interval $(-1/2,1/2]$ with addition modulo one. It is isomorphic to the unit circle in the complex plane, with the ordinary product of complex numbers. Let ${\mathbb{T}}_{+}=:[-1/4,1/4]\subseteq \mathbb{T}$. For a topological abelian group G, the character group or dual group ${G}^{\wedge}$ is the set of all continuous homomorphisms from G to $\mathbb{T}$, with addition defined pointwise.

## 2. The Connections between the Compact Covering Number and the d-Rank of Topological Groups

**Example 2.1.**Consider the group $G={\u2a01}_{n}{G}_{n}$, where ${G}_{n}={\u2a01}_{{\aleph}_{n}}\mathbb{Q}$ is discrete and G carries the product topology. A base of neighborhoods of zero is formed by the open subgroups ${W}_{m}={\u2a01}_{n>m}{G}_{n}$, so that G is r-disconnected as $r(G/{W}_{m})={\aleph}_{m}$ for every $m\in \mathbb{N}$. Moreover, $\varrho \left(G\right)=\left|G\right|={\aleph}_{\omega}$. On the other hand, every open subgroup H of G contains some ${W}_{m}$, so $r(G/H)\le r(G/{W}_{m})={\aleph}_{m}<\varrho \left(G\right)$.

**Fact 2.2.**Let G be an infinite abelian group. Then, the following statements are equivalent:

- (a)
- $r\left(G\right)<\infty $;
- (b)
- G is isomorphic to a subgroup of a group of the form ${\mathbb{Q}}^{m}\times {\prod}_{i=1}^{k}\mathbb{Z}\left({p}_{i}^{\infty}\right)$, where $m,k\in \mathbb{N}$ and ${p}_{1},\dots ,{p}_{k}$ are not necessarily distinct primes;
- (c)
- $G\cong L\times F\times {\prod}_{i=1}^{k}\mathbb{Z}\left({p}_{i}^{\infty}\right)$, where $k\in \mathbb{N}$, F is a finite abelian group, L is a subgroup of ${\mathbb{Q}}^{m}$ ($m\in \mathbb{N}$) and ${p}_{1},\dots ,{p}_{k}$ are not necessarily distinct primes;
- (d)
- G contains no infinite direct sum of non-trivial subgroups;
- (e)
- G contains no subgroup H, which is a direct sum of $\left|G\right|$-many non-trivial subgroups.

**Proof.**(a) → (b). Let $D\left(G\right)$ be the divisible hull of G. Then, $r\left(D\right(G\left)\right)=r\left(G\right)$, since ${r}_{0}\left(D\left(G\right)\right)={r}_{0}\left(G\right)$ and ${r}_{p}\left(D\left(G\right)\right)={r}_{p}\left(G\right)$ by the fact that G is essential in $D\left(G\right)$ (see [17] (Lemma 24.3)). Therefore, (a) implies that $r\left(D\right(G\left)\right)<\infty $. Now, (b) follows from the structure theorem for divisible abelian groups.

**Example 2.3.**For $\pi \subseteq \mathbb{P}$, one has $r\left({\mathbb{Q}}_{\pi}\right)=1$ and $\varrho \left({\mathbb{Q}}_{\pi}\right)=\left|\pi \right|$. In particular, $\varrho \left({\mathbb{Q}}_{\pi}^{n}\right)<\infty $ (for $n\in \mathbb{N}$) if and only if π is finite. Actually, a torsion-free group G has finite d-rank if and only if G is isomorphic to a subgroup of ${\mathbb{Q}}_{\pi}^{n}$ for some $n\in \mathbb{N}$ and some finite $\pi \subseteq \mathbb{P}$ ([18] (Lemma 10.8)).

**Fact 2.4.**([18] (Lemma 10.12)) A discrete abelian group G is non-r-disconnected (i.e., has $\varrho \left(G\right)<\infty $) if and only if $G\cong L\times F\times {\prod}_{i=1}^{k}\mathbb{Z}\left({p}_{i}^{\infty}\right)$, where L is a torsion-free non-r-disconnected group (i.e., isomorphic to a subgroup of ${\mathbb{Q}}_{\pi}^{n}$ for some finite $\pi \subseteq \mathbb{P}$ and $n\in \mathbb{N}$), F is a finite abelian group, $k\in \mathbb{N}$ and ${p}_{1},\dots ,{p}_{k}$ are not necessarily distinct primes.

**Fact 2.5.**According to the structure theory, an LCA group G is topologically isomorphic to ${\mathbb{R}}^{n}\times H$, where $n\in \mathbb{N}$ and the group H has a compact open subgroup K ([19]). Therefore, the quotient group $D=H/K$ is discrete.

**Example 2.6.**Let G be a non-r-disconnected LCA group.

**Proof.**(a) Indeed, as we saw above, there exists a closed subgroup H of G with a compact open subgroup K, such that $G={\mathbb{R}}^{n}\times H$, so $N={\mathbb{R}}^{n}\times K$ is an open subgroup of G. By our hypothesis, $H/K\cong G/N$ has finite d-rank. By Fact 2.4, $H/K\cong L\times F\times {\prod}_{i=1}^{k}\mathbb{Z}\left({p}_{i}^{\infty}\right)$, where L is a a torsion-free discrete non-r-disconnected group, F is a finite group and ${p}_{1},\dots ,{p}_{k}$, with $k\in \mathbb{N}$, are not necessarily distinct primes. Choosing K a bit larger, we can assume without loss of generality that $F=0$. Since $G/K\cong {\mathbb{R}}^{n}\times H/K$, we are done.

**Theorem 2.7.**Let G be an r-disconnected group. Then, $k\left(G\right)\ge \varrho \left(G\right)$. If G is LCA, then:

- (a)
- G has an open σ-compact subgroup L with $r(G/L)=\varrho \left(G\right)$.
- (b)
- every discrete quotient of G has a size at most $\varrho \left(G\right)$.
- (c)
- there exists a compact subgroup N of G, such that $G/N\cong {\mathbb{R}}^{n}\times D$ for some discrete abelian group D with $\left|D\right|=\varrho \left(G\right)$ and $n\in \mathbb{N}$.
- (d)
- $k\left(G\right)=\varrho \left(G\right)$.

**Proof.**Assume that $G={\bigcup}_{i\in I}{K}_{i}$, where each ${K}_{i}$ is compact. Then, for every open subgroup H of G with infinite $G/H$, the quotient map $q:G\to G/H$ takes each ${K}_{i}$ to a finite subset of $G/H$. Since $G/H$ is infinite, one has $|G/H|\le \left|I\right|$. This proves that $\varrho \left(G\right)\le k\left(G\right)$.

## 3. Some General Properties of Compatible Topologies

**Proposition 3.1.**If $(G,\tau )$ is Mackey, then:

**Proof.**(a) For the first inequality, it suffices to note that any compatible group topology is a subset of τ. For the second assertion, note that $|\tau |\le {2}^{\mathrm{w}\left(G\right)}$.

**Proposition 3.2.**Let $G={H}_{1}\times {H}_{2}$ be a locally quasi-convex group. Then,

**Proof.**This mapping is injective and preserves the order. It remains to show that it is well defined. Therefore, let τ be the original topology on G and ${\tau}_{1}$, ${\tau}_{2}$ the induced topologies on ${H}_{1}$ and ${H}_{2}$ respectively. Take ${\nu}_{1}\in \mathcal{C}({H}_{1},{\tau}_{1})$ and ${\nu}_{2}\in \mathcal{C}({H}_{2},{\tau}_{2})$. Then, ${\nu}_{1}\times {\nu}_{2}$ is again a locally quasi-convex topology on $G={H}_{1}\times {H}_{2}$. Clearly,

- dually closed if for every $x\in G\backslash H$, there exists $\chi \in {G}^{\wedge}$, such that $\chi \left(H\right)=\left\{0\right\}$ and $\chi \left(x\right)\ne 0$;
- dually embedded if each continuous character of H can be extended to a continuous character of G.

**Remark 3.3.**It is well known that if H is an open subgroup, it is dually closed and dually embedded, but in general, a closed subgroup need not have these properties.

- (a)
- It is easy to see that a subgroup H of a topological group $(G,\tau )$ is dually closed if and only if H is ${\tau}^{+}$-closed.
- (b)
- It is straightforward to prove that if $G={H}_{1}\oplus {H}_{2}$ is equipped with the product topology, then ${H}_{1}$ and ${H}_{2}$ are dually embedded in G. Moreover, ${H}_{1}$ and ${H}_{2}$ are dually closed in G precisely when ${H}_{1}$ and ${H}_{2}$ are maximally almost periodic.
- (c)
- It follows from Item (a) that a pair of compatible topologies shares the same dually-closed subgroups.

**Lemma 3.4.**Let $(G,\tau )$ be a topological abelian group and H a dually-closed and dually-embedded subgroup. If $M\subseteq H$ is quasi-convex in H, it is also quasi-convex in G.

**Proof.**We must check that every $x\in G\backslash M$ can be separated from M by means of a continuous character. Consider two cases:

**Proposition 3.5.**[20] (1.4) Every compact subgroup of a maximally almost periodic group is dually embedded and dually closed.

**Proposition 3.6.**Let $(G,\tau )$ be a locally quasi-convex group and H an open subgroup. Then, there exists a canonical poset embedding:

**Proof.**Fix $(H,\nu )\in \mathcal{C}(H,\tau {|}_{H})$, and let ${\mathcal{N}}_{H,\nu}\left(0\right)$ be a basis of quasi-convex zero neighborhoods for ν. If ${\mathcal{N}}_{H,\nu}\left(0\right)$ is considered as a basis of zero neighborhoods in G, we obtain a new group topology $\Psi \left(\nu \right)$ on G, for which H is an open subgroup. According to 3.4, $\Psi \left(\nu \right)$ is locally quasi-convex.

**Remark 3.7.**In the sequel, we denote by $Q(\tau )$ the quotient topology of a topology τ (the quotient group will always be quite clear from the context).

**Lemma 3.8.**Let $(G,\tau )$ be a MAP abelian group and H a dually-closed subgroup. Then, for the quotient $G/H$, one has $Q{(\tau )}^{+}=Q\left({\tau}^{+}\right)$, i.e., the Bohr topology of the quotient coincides with the quotient topology of the Bohr topology ${\tau}^{+}$.

**Proof.**Since both $Q{(\tau )}^{+}$ and $Q\left({\tau}^{+}\right)$ are precompact group topologies on $G/H$, it is sufficient to check that they are compatible. To this end, note that $Q{(\tau )}^{+}\ge Q\left({\tau}^{+}\right)$, since $Q(\tau )\ge Q\left({\tau}^{+}\right)$ and $Q\left({\tau}^{+}\right)$ is precompact. To see that they are compatible, take a $Q{(\tau )}^{+}$-continuous character $\chi :G/H\to \mathbb{T}$. Then, it is also $Q(\tau )$-continuous. Hence, the composition with the canonical projection $q:G\to G/H$ produces a τ-continuous character $\xi =\chi \circ q$ of G. Since ξ is ${\tau}^{+}$-continuous, as well, from the factorization $\xi =\chi \circ q$, we deduce that χ is also $Q\left({\tau}^{+}\right)$-continuous. Hence, $Q{(\tau )}^{+}\le Q\left({\tau}^{+}\right)$. ☐

**Notation 3.9.**Let H be a closed subgroup of the topological abelian group $(G,\tau )$. Denote by $q:G\to G/H$ the canonical projection. Further, for a group topology $\theta \in \mathcal{T}(G/H)$, we denote by ${q}^{-1}(\theta )$ the initial topology, namely the group topology $\{{q}^{-1}\left(O\right):\phantom{\rule{4pt}{0ex}}O\in \theta \}$. It is straightforward to prove that whenever θ is locally quasi-convex, then ${q}^{-1}(\theta )$ is locally quasi-convex, as well.

**Theorem 3.10.**Let H be a dually-closed subgroup of the locally quasi-convex Hausdorff group $(G,\tau )$. The mapping:

**Proof.**According to 3.9, $\Theta (\theta )$ is a locally quasi-convex group topology on G and finer than ${\tau}^{+}$, hence Hausdorff.

**Corollary 3.11.**Let G be a locally quasi-convex group, and let H be an open or a compact subgroup of G. Then, $\Theta :\mathcal{C}(G/H)\to \mathcal{C}\left(G\right)$ is a poset embedding.

**Remark 3.12.**Let $(G,\tau )$ be a locally quasi-convex group, and let H be an open subgroup of G. The images $\Theta \left(\mathcal{C}\right(G/H\left)\right)$ and $\Psi \left(\mathcal{C}\right(H\left)\right)$ in $\mathcal{C}\left(G\right)$ of both embeddings, obtained in Theorem 3.6 and Corollary 3.11, meet in a singleton, namely:

**Theorem 3.13.**Let $(G,\tau )$ be a locally quasi-convex Hausdorff group, and let K be a subgroup of G.

**Lemma 3.14 (Merzon).**Let ${\tau}_{1}\le {\tau}_{2}$ be group topologies on a group G, and let H be a subgroup of G. If the subspace topologies ${\tau}_{1}{{|}_{H}={\tau}_{2}|}_{H}$ and the quotient topologies $Q\left({\tau}_{1}\right)=Q\left({\tau}_{2}\right)$ coincide, then ${\tau}_{1}={\tau}_{2}$.

**Theorem 3.15.**Let $(G,\tau )$ be a locally quasi-convex Hausdorff group and K a compact subgroup of G. Then:

**Proof.**Let us show first that $Q:\mathcal{C}\left(G\right)\to \mathcal{C}(G/K),\theta \mapsto Q(\theta )$ is well defined. According to 3.13(b), $Q(\theta )$ is a locally quasi-convex Hausdorff group topology. In order to show that $Q(\theta )$ is compatible for $\theta \in \mathcal{C}\left(G\right)$, we fix a continuous character $\chi :(G/K,Q(\theta \left)\right)\to \mathbb{T}$. Let $q:G\to G/K$ be the quotient homomorphism. Then, $\chi \circ q:(G,\theta )\to \mathbb{T}$ is continuous. Since θ is compatible for $(G,\tau )$, also $\chi \circ q:(G,\tau )\to \mathbb{T}$ is continuous, and hence, $\chi :(G,Q(\tau \left)\right)\to \mathbb{T}$ is continuous. On the other hand, $\theta \ge {\tau}^{+}$, and hence, $Q(\theta )\ge Q\left({\tau}^{+}\right)=Q{(\tau )}^{+}$. Combining both conclusions, we obtain that $Q(\theta )$ is compatible for $(G,Q(\tau \left)\right)$.

**Corollary 3.16.**Let G be a locally compact abelian group and K a compact subgroup of G. Then, $\mathcal{C}\left(G\right)\phantom{\rule{4pt}{0ex}}\cong \mathcal{C}(G/K)$.

**Corollary 3.17.**Let G be a σ-compact group with an open compact subgroup. Then, there is a poset embedding $\mathcal{C}\left(G\right)\hookrightarrow {\mathfrak{F}}_{\omega}$; in particular, $\left|\mathcal{C}\left(G\right)\right|\le {2}^{\mathfrak{c}}$.

**Proof.**Let K be an open compact subgroup of G. Without loss of generality, we may assume that G is not compact, and hence, $G/K$ is infinite.

## 4. Compatible Topologies for Discrete Abelian Groups

**Notation 4.1.**Every free filter φ on γ defines a topology ${\tau}_{\phi}$ on G with a base $\{{W}_{B}:B\in \phi \}$ of neighborhoods of zero, where:

**Lemma 4.2.**Let δ denote the discrete topology on G. The mapping:

**Proof.**Obviously ${\delta}^{+}\le {\tau}_{\phi}\vee {\delta}^{+}\le \delta $, so the topology ${\tau}_{\phi}\vee {\delta}^{+}$ is compatible. Since ${\delta}^{+}$ is precompact, ${\delta}^{+}$ is also locally quasi-convex. Moreover, ${\tau}_{\phi}\vee {\delta}^{+}$ is locally quasi-convex, as proven in Notation 4.1. This shows that Ξ is well defined.

**Remark 4.3.**Observe that $\Xi \left(\mathcal{P}(\gamma )\right)=\delta $.

**Corollary 4.4.**For G as above, the sets $\mathcal{C}\left(G\right)$ and ${\mathfrak{F}}_{\gamma}$ are quasi-isomorphic, in particular $\mathrm{width}\phantom{\rule{0.166667em}{0ex}}\left(\mathcal{C}\left(G\right)\right)=\left|\mathcal{C}\left(G\right)\right|={2}^{{2}^{\left|G\right|}}.$

**Proof.**According to 4.2 and 1.6, the sets $\mathcal{C}\left(G\right)$ and ${\mathfrak{F}}_{\gamma}$ are quasi-isomorphic. Since $\left|G\right|=\gamma $ and since there are ${2}^{{2}^{\gamma}}$ different free ultrafilters in ${\mathfrak{F}}_{\gamma}$, the first assertion follows.

**Theorem 4.5.**Let G be a discrete abelian group of infinite rank. Then, $\mathcal{C}\left(G\right)\stackrel{q.i.}{\phantom{\rule{0.0pt}{0ex}}\cong}{\mathfrak{F}}_{\left|G\right|}$ holds. In particular, G admits ${2}^{{2}^{\left|G\right|}}$ pairwise incomparable compatible group topologies and $\left|\mathcal{C}\left(G\right)\right|={2}^{{2}^{\left|G\right|}}$.

**Proof.**Let $\gamma =\left|G\right|$. According to 1.6, there are poset embeddings:

**Corollary 4.6.**Suppose that H is a discrete abelian group for which ${\mathfrak{F}}_{\omega}$ does not embed in $\mathcal{C}\left(H\right)$. Then, H has finite d-rank; in particular, H is countable.

**Proof.**If H has infinite d-rank, then 4.5 implies the existence of a poset embedding ${\mathfrak{F}}_{\left|H\right|}\to \mathcal{C}\left(H\right)$. Hence, the hypothesis implies that H has finite rank. ☐

## 5. Proofs of Theorem A, Theorem B and Corollary C

**Corollary 5.1.**If G is an LCA group, such that ${\mathfrak{F}}_{\omega}$ does not embed in $\mathcal{C}\left(G\right)$, then G is non-r-disconnected (so G contains a compact subgroup K, such that $G/K\cong {\mathbb{R}}^{n}\times L\times {\prod}_{i=1}^{k}\mathbb{Z}\left({p}_{i}^{\infty}\right)$ for some $n,m,k\in \mathbb{N}$, a subgroup L of ${\mathbb{Q}}^{m}$ and not necessarily distinct primes ${p}_{i}$).

**Proof.**This follows from Example 2.6 and Theorem A. ☐

## 6. Metrizable Separable Mackey Groups with Many Compatible Topologies

**Theorem 6.1.**[23] The group $G=({c}_{0}\left(X\right),{\mathfrak{u}}_{0})$ is a non-compact Polish connected Mackey group with ${\mathfrak{u}}_{0}^{+}={\mathfrak{p}}_{0}$.

**Notation 6.2.**For a subset B of $\mathbb{N}$ and a neighborhood U of 0 in X, let:

- (a)
- ${\bigcap}_{i\in I}P({B}_{i},U)=P({\bigcup}_{i\in I}{B}_{i},U)$ for every subset $\{{B}_{i}:i\in I\}\subseteq \mathcal{P}\left(\mathbb{N}\right)$;
- (b)
- if $U\ne X$ and $P({B}_{1},{U}_{1})\subseteq P(B,U)$, then $B\subseteq {B}_{1}$.

**Definition 6.3.**For any $A\subseteq \mathbb{N}$, define a group topology ${\mathfrak{t}}_{A}$ on ${c}_{0}\left(X\right)$ having as a neighborhood basis at zero the family of sets $\left(P\right(B,U\left)\right)$, where U runs through all neighborhoods of zero in X and B through all elements of $\mathcal{P}\left(\mathbb{N}\right)$ with finite $A\Delta B$.

**Lemma 6.4.**If $P(A,U)\in {\mathfrak{t}}_{B}$ with $U\ne X$, then $A\backslash B$ is finite. Consequently, $A\backslash B$ is finite whenever ${\mathfrak{t}}_{A}\le {\mathfrak{t}}_{B}$.

**Proof.**By our hypothesis, there exists a subset ${B}^{\prime}$ of $\mathbb{N}$, such that ${B}^{\prime}\Delta B$ is finite and $P({B}^{\prime},{U}^{\prime})\subseteq P(A,U)$ for some neighborhood ${U}^{\prime}$ of 0 in X. Then, by Item (b) of Notation 6.2, $A\subseteq {B}^{\prime}$. Since $|{B}^{\prime}\Delta B|<\infty $, this proves that $A\backslash B$ is finite, as well.

**Proposition 6.5.**For the topological group $(G,{\mathfrak{u}}_{0})=({c}_{0}\left(X\right),{\mathfrak{u}}_{0})$, the mapping:

**Proof.**The mapping (7) is well defined, since for any $A\subseteq \mathbb{N}$, we have ${\mathfrak{p}}_{0}\subseteq {\mathfrak{t}}_{A}\subseteq {\mathfrak{u}}_{0}$. Since all of the topologies ${\mathfrak{t}}_{A}$ are also locally quasi-convex, they belong to $\mathcal{C}\left(G\right)$.

**Corollary 6.6.**The mapping $\mathcal{P}{\left(\mathbb{N}\right)}_{*}\to \mathcal{C}\left(G\right),\phantom{\rule{4pt}{0ex}}{A}_{*}\mapsto {\mathfrak{t}}_{A}$ is a poset embedding.

**Remark 6.7.**The above Proposition 6.5 implies that $sup\{{\mathfrak{t}}_{A},{\mathfrak{t}}_{B}\}\le {\mathfrak{t}}_{A\cup B}$. However, one can easily check with Notation 6.2(a) that actually, $sup\{{\mathfrak{t}}_{A},{\mathfrak{t}}_{B}\}={\mathfrak{t}}_{A\cup B}$ holds true. In particular, if $P(A,U)\in sup\{{\mathfrak{t}}_{{B}_{1}},\dots ,{\mathfrak{t}}_{{B}_{n}}\}={\mathfrak{t}}_{{B}_{1}\cup \dots \cup {B}_{n}}$ with $U\ne X$, then $A{\subseteq}^{*}{\bigcup}_{i}{B}_{i}$ by Lemma 6.4.

**Example 6.8.**There exists an anti-chain of size $\mathfrak{c}$ in $(\mathcal{P}{\left(\mathbb{N}\right)}_{*},\subseteq )$. Although this fact is well known (see, for example, [24]), we give a brief argument for the reader’s convenience.

**Theorem 6.9.**The poset $\mathcal{C}\left(G\right)$ is quasi-isomorphic to $\mathcal{P}\left(\mathbb{R}\right)$, so $\left|\mathcal{C}\left(G\right)\right|={2}^{\mathfrak{c}}$ and $\mathcal{C}\left(G\right)$ has chains of length ${\mathfrak{c}}^{+}$.

**Proof.**According to Proposition 3.1, $\mathcal{C}\left(G\right)$ embeds into $\mathcal{P}\left(\mathbb{R}\right)$ as $(G,{\mathfrak{u}}_{0})$ is Mackey and $|{\mathfrak{u}}_{0}|=\mathfrak{c}$. Therefore, it suffices to show that $\mathcal{P}\left(\mathbb{R}\right)$ embeds into $\mathcal{C}\left(G\right)$. It is enough to produce an embedding of $\mathcal{P}\left(\mathbb{R}\right)\backslash \left\{\varnothing \right\}$ into $\mathcal{C}\left(G\right)\backslash \left\{{\mathfrak{p}}_{0}\right\}$. To this end, we shall replace $\mathbb{R}$ by an almost disjoint family $\mathfrak{A}$ of size $\mathfrak{c}$ of infinite subsets of $\mathbb{N}$ (see Example 6.8).

**Example 6.10.**Let $\mathfrak{A}$ be as in the above proof. We show that ${\mathcal{T}}_{\mathcal{B}}$ is not metrizable whenever $\mathcal{B}$ is an uncountable subset of $\mathfrak{A}$. Since there are ${2}^{\mathfrak{c}}$ many such sets, this provides ${2}^{\mathfrak{c}}$ many non-metrizable topologies in $\mathcal{C}\left(G\right)$.

## 7. Final Comments and Open Questions

**Questions 7.1.**

- (a)
- Compute $\left|\mathcal{C}\right(\mathbb{Z}\left)\right|$. Is it infinite? Is it countable? Is it at most $\mathfrak{c}$? At least $\mathfrak{c}$?
- (b)
- Compute $\left|\mathcal{C}\right(\mathbb{R}\left)\right|$. Is it infinite? Is it countable? Is it at most $\mathfrak{c}$? At least $\mathfrak{c}$?
- (c)
- Compute $\left|\mathcal{C}\right(\mathbb{Z}\left({p}^{\infty}\right)\left)\right|$, where p is a prime. Is it infinite? Is it countable? Is it at most $\mathfrak{c}$? At least $\mathfrak{c}$?

**Question 7.2.**Is $\mathcal{C}(\mathbb{R}\oplus {\u2a01}_{{\omega}_{1}}{\mathbb{Z}}_{2})\stackrel{q.i.}{\phantom{\rule{0.0pt}{0ex}}\cong}{\mathfrak{F}}_{{\omega}_{1}}$?

**Question 7.3.**Let G be a non-precompact second countable Mackey group. Is it true that $\left|\mathcal{C}\right(G\left)\right|\ge \mathfrak{c}$?

**Problem 7.4.**Find sufficient conditions for a metrizable precompact group G to be Mackey (i.e., have $\left|\mathcal{C}\right(G\left)\right|=1$).

**Question 7.5.**If for a group G, one has $\left|\mathcal{C}\right(G\left)\right|>1$, can $\mathcal{C}\left(G\right)$ be finite? In particular, can one have a Mackey group G with $\left|\mathcal{C}\right(G\left)\right|=2$?

**Conjecture 7.6.**[Mackey dichotomy] For a locally quasi-convex group G, one has either $\left|\mathcal{C}\right(G\left)\right|=1$ or $\left|\mathcal{C}\right(G\left)\right|\ge \mathfrak{c}$.

**Question 7.7.**How large can $\mathcal{C}(\mathbb{Z},\lambda )$ be?

**Question 7.8.**How large can be $\mathcal{C}(G,\tau )$ in this case?

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Varopoulos, N.T. Studies in harmonic analysis. Proc. Camb. Phil. Soc.
**1964**, 60, 467–516. [Google Scholar] [CrossRef] - Chasco, M.J.; Martín-Peinador, E.; Tarieladze, V. On Mackey topology for groups. Stud. Math.
**1999**, 132, 257–284. [Google Scholar] - De Leo, L. Weak and Strong Topologies in Topological Abelian Groups. Ph.D. Thesis, Universidad Complutense de Madrid, Madrid, Spain, July 2008. [Google Scholar]
- De Leo, L.; Dikranjan, D.; Martín-Peinador, E.; Tarieladze, V. Duality Theory for Groups Revisited: g-barrelled groups, Mackey & Arens Groups. 2015; in preparation. [Google Scholar]
- Bonales, G.; Trigos-Arrieta, F.J.; Mendoza, R.V. A Mackey-Arens theorem for topological Abelian groups. Bol. Soc. Mat. Mex. III
**2003**, 9, 79–88. [Google Scholar] - Berarducci, A.; Dikranjan, D.; Forti, M.; Watson, S. Cardinal invariants and independence results in the lattice of precompact group topologies. J. Pure Appl.
**1998**, 126, 19–49. [Google Scholar] [CrossRef] - Comfort, W.; Remus, D. Long chains of Hausdorff topological group topologies. J. Pure Appl. Algebra
**1991**, 70, 53–72. [Google Scholar] [CrossRef] - Comfort, W.; Remus, D. Long chains of topological group topologies—A continuation. Topology Appl.
**1997**, 75, 51–79. [Google Scholar] [CrossRef] - Dikranjan, D. The Lattice of Compact Representations of an infinite group. In Proceedings of Groups 93, Galway/St Andrews Conference; London Math. Soc. Lecture Notes 211. Cambidge Univ. Press: Cambridge, UK, 1995; pp. 138–155. [Google Scholar]
- Dikranjan, D. On the poset of precompact group topologies. In Topology with Applications, Proceedings of the 1993 Szekszàrd (Hungary) Conference; Bolyai Society Mathematical Studies. Czászár, Á., Ed.; Elsevier: Amsterdam, The Netherlands, 1995; Volume 4, pp. 135–149. [Google Scholar]
- Dikranjan, D. Chains of pseudocompact group topologies. J. Pure Appl. Algebra
**1998**, 124, 65–100. [Google Scholar] [CrossRef] - Engelking, R. General Topology(Sigma Series in Pure Mathematics, 6), 2nd ed.; Heldermann Verlag: Berlin, Germany, 1989. [Google Scholar]
- Abramsky, S.; Jung, A. Domain theory. In Handbook of Logic in Computer Science III; Abramsky, S., Gabbay, D.M., Maibaum, T.S.E., Eds.; Oxford University Press: New York, NY, USA, 1994; pp. 1–168. [Google Scholar]
- Banaszczyk, W. Additive Subgroups of Topological Vector Spaces; Lecture Notes in Mathematics; Springer Verlag: Berlin, Germany, 1991; Volume 1466. [Google Scholar]
- Enflo, P. Uniform structures and square roots in topological groups. Israel J. Math.
**1970**, 8, 230–252. [Google Scholar] [CrossRef] - Außenhofer, L.; Dikranjan, D.; Martín-Peinador, E. Locally quasi-convex compatible topologies on σ-compact LCA groups. 2015; in preparation. [Google Scholar]
- Fuchs, L. Infinite Abelian Groups; Academic Press: New York, NY, USA, 1970. [Google Scholar]
- Dikranjan, D.; Shakhmatov, D. Topological groups with many small subgroups. Topology Appl.
**2015**, in press. [Google Scholar] - Dikranjan, D.; Prodanov, I.; Stojanov, L. Topological Groups (Characters, Dualities, and Minimal Group Topologies); Marcel Dekker, Inc.: New York, NY, USA, 1990. [Google Scholar]
- Bruguera, M.; Martín-Peinador, E. Open subgroups, compact subgroups and Binz-Butzmann reflexivity. Topology Appl.
**1996**, 72, 101–111. [Google Scholar] [CrossRef] - Außenhofer, L. A note on weakly compact subgroups of locally quasi-convex groups. Arch. Math.
**2013**, 101, 531–540. [Google Scholar] [CrossRef] - Dikranjan, D.; Protasov, I. Counting maximal topologies on countable groups and rings. Topology Appl.
**2008**, 156, 322–325. [Google Scholar] [CrossRef] - Dikranjan, D.; Martín-Peinador, E.; Tarieladze, V. Group valued null sequences and metrizable non-Mackey groups. Forum Math.
**2014**, 26, 723–757. [Google Scholar] [CrossRef] - Sierpinski, W. Cardinal and ordinal numbers; Panstwowe Wydawnictwo Naukowe: Warsaw, Poland, 1958. [Google Scholar]
- Baumgartner, J.E. Almost disjoint sets, the dense set problem and the partition calculus. Ann. Math. Logic
**1976**, 10, 401–439. [Google Scholar] [CrossRef] - De la Barrera Mayoral, D.; Dikranjan, D.; Martìn Peinador, E. “Varopoulos paradigm”: Mackey property vs. metrizability in topological groups. 2015; in preparation. [Google Scholar]
- Außenhofer, L.; de la Barrera Mayoral, D. Linear topologies on ℤ are not Mackey topologies. J. Pure Appl. Algebra
**2012**, 216, 1340–1347. [Google Scholar] [CrossRef] - De la Barrera Mayoral, D. ℚ is not Mackey group. Topology Appl.
**2014**, 178, 265–275. [Google Scholar] [CrossRef]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Außenhofer, L.; Dikranjan, D.; Martín-Peinador, E.
Locally Quasi-Convex Compatible Topologies on a Topological Group. *Axioms* **2015**, *4*, 436-458.
https://doi.org/10.3390/axioms4040436

**AMA Style**

Außenhofer L, Dikranjan D, Martín-Peinador E.
Locally Quasi-Convex Compatible Topologies on a Topological Group. *Axioms*. 2015; 4(4):436-458.
https://doi.org/10.3390/axioms4040436

**Chicago/Turabian Style**

Außenhofer, Lydia, Dikran Dikranjan, and Elena Martín-Peinador.
2015. "Locally Quasi-Convex Compatible Topologies on a Topological Group" *Axioms* 4, no. 4: 436-458.
https://doi.org/10.3390/axioms4040436